Make the roux:
Adjust oven rack to lowest position and heat oven to 350°F. Toast 1/2 cup flour in Dutch oven on stovetop over medium heat, stirring constantly, until just beginning to brown, about 5 minutes. Off heat, whisk in oil until smooth. Cover, transfer pot to oven, and cook until mixture is deep brown and fragrant, about 45 minutes. (Roux can be refrigerated in airtight container for 1 week. To use, heat in Dutch oven over medium-high heat, whisking constantly, until just smoking, and continue with step 2.)
Transfer Dutch oven to stovetop and whisk cooked roux to combine. Add onion, bell pepper, and celery and cook over medium heat, stirring frequently, until softened, about 10 minutes. Stir in remaining 1 tablespoon flour, garlic, thyme, and cayenne and cook until fragrant, about 1 minute. Add tomatoes and cook until dry, about 1 minute. Slowly whisk in broth and fish sauce until smooth. Season chicken with pepper. Add chicken and bring to boil.
Reduce heat to medium-low and simmer, covered, until the chicken is tender and reaches 165°F (74°C) in the thickest part, about 30 minutes. Skim fat and transfer chicken to plate. When chicken is cool enough to handle, cut into bite-size pieces and return to pot; discard bones.
Stir in sausage and okra, if using, and simmer until heated through, about 5 minutes. Add shrimp (split lengthwise) and simmer until cooked through, about 5 minutes. Season with salt and pepper. Serve. (Gumbo can be refrigerated in airtight container for 1 day.)
MAKE AHEAD: Gumbo can be made through step 3 and refrigerated in an airtight container for 3 days. To serve, bring gumbo to simmer, covered, in Dutch oven. Remove lid and proceed with recipe as directed.